# Relationship between trace and phase space

1. Jun 18, 2013

### sunrah

So I noticed we can define entropy in two very different ways:
1) quantum mechanically
$S = -k Tr(\hat{\rho}\ln{(\hat{\rho})})$
2) classically
$S = -k \int \rho \ln{(\rho)} d\Gamma$

where Tr is the trace and $d\Gamma = \frac{1}{h^{3N}N!}\prod_{i}^{N} d^{3N}q_{i}d^{3N}p_{i}$ is the phase space volume element.

My question is how summing over the diagonal of an operator is like integrating the same quantity over phase space.

Edit: I realise that we are working out a mean value.

Last edited: Jun 18, 2013